#7626834

order	512
nilpotency class	2
isoclinism	isoclinic class of #62892
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁶
commutator	C₂²	C₂⁵×C₄
prime radical	C₂×#3459	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6940 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 36 points:

(1,2,4,3)(5,6,11,15)(7,14,18,12)(8,10,9,13)(16,17,19,20)(21,22,28,26)(23,29,34,33)(24,35,25,30)(27,36,32,31);(5,7)(6,12)(8,19)(9,16)(10,17)(11,18)(13,20)(14,15)(21,23,28,34)(22,29,26,33)(24,32,25,27)(30,36,35,31);(1,3,4,2)(5,8,11,9)(6,13,15,10)(7,16,18,19)(12,17,14,20)(21,24,28,25)(22,30,26,35)(23,32,34,27)(29,36,33,31);(5,9,11,8)(6,10,15,13)(7,16,18,19)(12,17,14,20)(21,25,28,24)(22,30,26,35)(23,27,34,32)(29,36,33,31);(5,7)(6,14)(8,16)(9,19)(10,17)(11,18)(12,15)(13,20)(21,26,28,22)(23,33,34,29)(24,35,25,30)(27,36,32,31);(5,10)(6,8)(7,17)(9,15)(11,13)(12,19)(14,16)(18,20)(21,27)(22,31)(23,24)(25,34)(26,36)(28,32)(29,30)(33,35)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=i,d^2=h,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^d=eh

Conjugacy classes & character table

146 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534030);